
Namely, in this case, these couplings do not depend on the masses of the additional Higgs bosons and the soft breaking masses, and so, the constraints from the RG evolution can be evaded.
We also note that a further requirement, \(\lambda _{10}=0\)  leads to \(\lambda _7=\lambda _8=\lambda _9=0\)  and \(\lambda _1=\lambda _2=\lambda _3=2\lambda _4=2\lambda _5=2\lambda _6\) .
In this limit, the 3HDM potential \(V_{\rm 3HDM}\) 
except for the soft breaking terms \(V_{\rm soft}\) 
has an Sp(6) symmetry [1]}, which is explicitly broken by the terms with the coefficients \(M_{12}^2\)  or \(\kappa _{m \phi n}v^2_{S_m}\ (m=1,\bar{2},\ n=1,2,3)\) . In fact,
the allowed region for \(\lambda _{10}\)  is \(-0.2\lesssim \lambda _{10} \lesssim 0.2 \) , which contains this limit.
Hence, the Sp(6) symmetry would be the desired (approximate) symmetry for the 3HDM part of the Higgs potential to satisfy all the theoretical and experimental constraints.
